# New Physicochemical Methodology for the Determination of the Surface Thermodynamic Properties of Solid Particles

^{1}

^{2}

*AppliedChem*)

## Abstract

**:**

## 1. Introduction

_{2}- is equal to:

- 2.
- The method proposed by Schultz et al. [73], using the Fowkes relation [72], similarly gave the free energy of adsorption $\Delta {G}_{a}^{0}$ as a function of the geometric mean of the respective dispersive components of the surface energy of the liquid solvent ${\gamma}_{l}^{d}$ and the solid ${\gamma}_{s}^{d}$:

- 3.
- The method deduced from the works of Sawyer and Brookman [70] used:

- 4.
- The method of the deformation polarizability ${\alpha}_{0}$ proposed by Donnet et al. [80]. They proposed the following relation:

- 5.
- Chehimi et al. [59] used the standard enthalpy of vaporization $\Delta {H}_{vap.}^{0}$ (supposed constant) of n-alkanes and polar molecules:

- 6.
- The method of Brendlé and Papirer [2] used the concept of the topological index ${\chi}_{T}$; that is, a parameter considering the topology and the local electronic density in the polar probe structure. They gave the following relation:

_{A}and K

_{D}, of solids can be determined by Papirer following the relation [12,13]:

## 2. New Methodology

#### 2.1. Molecular Models

#### 2.2. Hamieh’s Thermal Model

^{2}and constant for any used temperature. Hamieh [69] proved the non-validity of the Dorris–Gray method and gave the following expression of ${a}_{-CH2-}$ (in Å

^{2}) as a function of the temperature T (in K):

#### 2.3. The New Lewis’s Acid Base Parameters

_{D}+ K

_{A}X

_{2}− K X

_{3}

_{1}, X

_{2}and X

_{3}are known for every polar molecule, whereas K

_{D}, K

_{A}and K are the unknown parameters. The problem, given by Equation (16), is represented by a linear system for the N solvents and admits a unique solution for N ≥ 3, giving the three unknown constants numbers: K

_{D}, K

_{A}and K.

## 3. Materials and Solvents

^{d}, was given by Riddle and Fowkes [83], who subtracted the contribution of the Van der Waals interactions (or dispersion forces). This acceptor number was normalized by Hamieh et al. [77,81], who proposed to use a dimensionless donor number DN′ and a dimensionless acceptor number AN′. All probes (Aldrich) were highly pure grade (i.e., 99%). The probes used were n-alkanes (pentane, hexane, heptane, octane and nonane); amphoteric solvents: acetonitrile, acetone; basic solvents: ethyl acetate, tetrahydrofuran (THF) and acidic solvents: chloroform and nitromethane.

_{R}, was used for the calculation. The standard deviation was less than 1% in all measurements. All columns used in this study were prepared using a stainless-steel column with a 2 mm inner diameter and with an approximate length of 20 cm.

## 4. Results

#### 4.1. Determination of the Gibbs Free Energy of Adsorption

_{2}Cl

_{2}> THF > CHCl

_{3}> Toluene > CCl

_{4}

#### 4.2. London Dispersive Surface Energy of Alumina Particles

- The group, constituted by the Kiselev, cylindrical, VDW, geometric and Doris–Gray models, taking into account the geometric form of n-alkanes, and they presented very close values of ${\gamma}_{s}^{d}$ and the surface of methylene group (Figure 3).
- The second group concerns the models relative to thermal model, Redlich–Kwong equation and the global average results that concluded to the more accurate values of the ${\gamma}_{s}^{d}$ of the alumina surfaces (Figure 3).

#### 4.3. Surface Thermodynamic of Alumina Particles

#### 4.3.1. The Gibbs Specific Free Energy of Adsorption

#### 4.3.2. Lewis’s Acid Base Parameters

_{4}, CH2Cl

_{2}, CHCl

_{3}, diethyl ether, THF and toluene on alumina particles, by using the various molecular models and methods. The results are presented on Table 8 and Table 9.

_{4}is 51.86%, followed by toluene (21.96%), CH

_{2}Cl

_{2}(20.47%), CHCl

_{3}(13.91%), THF (8.18%) and diethyl ether (5.01%). The results in Table 8 showed that the IGC methods that better match the thermal models are the following: boiling point, vapor pressure and enthalpy of vaporization; followed by the other molecular models, such as the cylindrical and Kiselev models.

_{4}and 31.97% with CH

_{2}Cl

_{2}, followed by THF (12.98%), CHCl

_{3}(12.46%), THF (8.18%) and diethyl ether (5.01%). The closer methods to the thermal models are identical to those obtained with the specific enthalpy of adsorption, proving the effect of the temperature on the surface area of the organic molecules.

_{4}. Table S1 proved the presence of a maximum of ${\mathsf{\Omega}}_{a}^{sp}$ for all models and IGC methods, followed by CH

_{2}Cl

_{2}and toluene by using the Hamieh model. This result again proved the strong basicity of the alumina particles. Indeed, the strong probability of more acidic solvents is the highest compared to other solvents.

## 5. Study of the Surface Properties of Other Oxides

#### 5.1. Case of TiO_{2} Particles

_{2}particles that exhibited a specific surface area of 59 m

^{2}/g. We used the values of the surface areas of the organic probes given as a function of the temperature given by relations (13)–(15) to calculate the London dispersive surface energy and the Lewis acid–base parameters. The obtained results are presented in Table 11.

_{2}particles, which is ten times more basic than acidic and also proved a decrease of the London dispersive surface energy from 79.8 $\mathrm{mJ}/{\mathrm{m}}^{2}$ to 41.0 $\mathrm{m}\mathrm{J}/{\mathrm{m}}^{2}$ in the temperature interval [40 °C; 120 °C].

#### 5.2. Case of SiO_{2} Particles

#### 5.3. Comparison between the Three Oxides

_{2}> Al

_{2}O

_{3}> TiO

_{2}

_{2}> Al

_{2}O

_{3}> TiO

_{2}

_{2}> Al

_{2}O

_{3}> SiO

_{2}

## 6. Conclusions

_{2}> Al

_{2}O

_{3}> TiO

_{2}

## Supplementary Materials

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Variations of the surface free energy ($-\Delta {G}_{a}^{0}\left(T\right)\left(in\mathrm{J}/\mathrm{mol}\right)$ of the solvents adsorbed on alumina particles as a function of the temperature T (K).

**Figure 2.**Dispersive component of the surface energy ${\gamma}_{s}^{d}\left(\mathrm{mJ}/{\mathrm{m}}^{2}\right)$ of alumina particles as a function of the temperature T (K) using different methods and models.

**Figure 3.**Dispersive surface energy ${\gamma}_{s}^{d}\left(\mathrm{mJ}/{\mathrm{m}}^{2}\right)$ of alumina particles following the various methods and models for eight temperatures.

**Figure 4.**Variations of $\Delta {G}_{a}^{sp}$ as a function of the temperature for the various polar molecules adsorbed on the alumina surface by using the different IGC models and methods.

**Figure 5.**Variations of $\left(\frac{-\Delta {H}_{a}^{sp}}{A{N}^{\u2019}}\right)$ as a function of $\left(\frac{D{N}^{\u2019}}{A{N}^{\u2019}}\right)$ of different polar molecules adsorbed on alumina surface for different molecular models and IGC methods.

**Figure 6.**Variations of $\left(\frac{-\Delta {S}_{a}^{sp}}{A{N}^{\u2019}}\right)$ as a function of $\left(\frac{D{N}^{\u2019}}{A{N}^{\u2019}}\right)$ of different polar molecules adsorbed on alumina particles for different molecular models and IGC methods.

**Table 1.**Surface areas of n-alkanes (in Å

^{2}) using the various molecular models: spherical (Sph.), geometric (Geom.), Redlich-Kwong (R-K), cylindric (Cyl.), Kiselev and Van der Waals (VDW).

Cn | Sph. | Geom. | R-K | Cyl. | Kiselev | VDW |
---|---|---|---|---|---|---|

C5 | 36.4 | 32.9 | 36.8 | 39.3 | 45 | 47 |

C6 | 39.6 | 40.7 | 41.3 | 45.5 | 51.5 | 52.7 |

C7 | 42.7 | 48.5 | 46.4 | 51.8 | 57 | 59.2 |

C8 | 45.7 | 56.2 | 50.8 | 58.1 | 63 | 64.9 |

C9 | 48.7 | 64 | 54.5 | 64.4 | 69 | 69.6 |

C10 | 51.7 | 71.8 | 58.2 | 70.7 | 75 | 74.4 |

Probes | DN′ | AN′ | DN′/AN′ | Acid Base Force |
---|---|---|---|---|

CCl_{4} | 0 | 2.3 | 0 | Acid |

CHCl_{3} | 0 | 18.7 | 0 | Stronger acidity |

CH_{2}Cl_{2} | 3 | 13.5 | 0.2 | Weaker amphoteric |

Toluene | 9.75 | 3.3 | 3.0 | Amphoteric |

Diethyl ether | 48 | 4.9 | 9.8 | Amphoteric |

THF | 50 | 1.9 | 26.3 | Stronger Basicity |

**Table 3.**Variations of the Gibbs free energy ($-\Delta {G}_{a}^{0}{\mathrm{in}\mathrm{J}\mathrm{mol}}^{-1}$) of adsorption of the various polar solvents on alumina particles as a function of the temperature.

T(K) | 303.15 | 323.15 | 343.15 | 363.15 | 383.15 | 403.15 | 423.15 | 443.15 | 463.15 |
---|---|---|---|---|---|---|---|---|---|

Pentane | 25,573 | 25,539 | 25,470 | 25,441 | 25,397 | 25,353 | 25,309 | 25,265 | 25,573 |

Hexane | 28,968 | 28,878 | 28,790 | 28,698 | 28,603 | 28,522 | 28,428 | 28,338 | 28,968 |

Heptane | 31,940 | 31,857 | 31,774 | 31,692 | 31,609 | 31,527 | 31,444 | 31,361 | 31,123 |

Octane | 35,420 | 35,117 | 34,813 | 34,510 | 34,207 | 33,904 | 33,601 | 33,604 | 32,995 |

Nonane | 38,821 | 38,467 | 37,716 | 37,163 | 36,611 | 36,058 | 35,506 | 34,953 | 34,401 |

CH_{2}Cl_{2} | 61,952 | 59,637 | 57,919 | 56,367 | 54,442 | 52,966 | 51,248 | 49,509 | 47,769 |

CHCl_{3} | 45,147 | 42,524 | 40,448 | 38,512 | 36,838 | 34,950 | 32,911 | 31,850 | 29,664 |

CCl_{4} | 34,479 | 34,514 | 34,449 | 34,435 | 34,420 | 34,405 | 34,391 | 34,376 | 34,361 |

THF | 64,519 | 62,228 | 60,464 | 58,838 | 57,449 | 55,865 | 54,281 | 53,324 | 51,507 |

Ether | 67,319 | 65,062 | 63,377 | 61,763 | 60,317 | 58,729 | 56,976 | 55,555 | 53,967 |

Toluene | 47,084 | 46,302 | 45,020 | 44,028 | 43,511 | 42,617 | 41,724 | 40,831 | 39,937 |

**Table 4.**Values of standard enthalpy ($-\Delta {H}_{a}^{0}\left({\mathrm{J}\mathrm{mol}}^{-1}\right)$) and entropy ($-\Delta {S}_{a}^{0}\left({\mathrm{J}\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}\right)$) of adsorption of the various organic molecules adsorbed on alumina surfaces.

Probes | $-\mathbf{\Delta}{\mathit{H}}_{\mathit{a}}^{0}$ | $-\mathbf{\Delta}{\mathit{S}}_{\mathit{a}}^{0}$ | $\mathbf{Equation}\text{}\mathbf{of}-\mathbf{\Delta}{\mathit{G}}_{\mathit{a}}^{0}\left(\mathit{T}\right)$ | R^{2} |
---|---|---|---|---|

Pentane | 26,284 | 2.2 | $-\Delta {G}_{a}^{0}\left(T\right)$= −2.2 T + 26,284 | 0.9967 |

Hexane | 30,423 | 4.5 | $-\Delta {G}_{a}^{0}\left(T\right)$= −4.5 T + 30,423 | 0.9967 |

Heptane | 33,192 | 4.1 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −4.1 T + 33,192 | 1.0000 |

Octane | 40,094 | 15.4 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −15.4 T + 40,094 | 0.9989 |

Nonane | 47,409 | 28.1 | $-\Delta {G}_{a}^{0}\left(T\right)$= −28.1 T + 47,409 | 0.9985 |

CCl_{4} | 34,696 | 0.7 | $-\Delta {G}_{a}^{0}\left(T\right)$= −0.7 T + 34,696 | 0.9991 |

CHCl_{3} | 72,931 | 93.8 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −93.8 T + 72,931 | 0.9949 |

CH_{2}Cl_{2} | 87,807 | 86.6 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −86.6 T + 87,807 | 0.9985 |

Toluene | 60,493 | 44.4 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −44.4 T + 60,493 | 0.9978 |

THF | 86,959 | 76.8 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −76.8 T + 86,959 | 0.9970 |

Ether | 91,555 | 81.5 | $-\Delta {G}_{a}^{0}\left(T\right)$ = −81.5 T + 91,555 | 0.9976 |

**Table 5.**Values of the dispersive component of the surface energy ${\gamma}_{s}^{d}\left(\mathrm{mJ}/{\mathrm{m}}^{2}\right)$ of alumina particles as a function of the temperature.

${\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathbf{m}\mathbf{J}/{\mathbf{m}}^{2}\right)\text{}\left(\mathbf{Alumina}\right)$ | ||||||||
---|---|---|---|---|---|---|---|---|

T (K) | 323.15 | 343.15 | 363.15 | 383.15 | 403.15 | 423.15 | 443.15 | 463.15 |

Kiselev | 53.0 | 47.1 | 41.7 | 37.8 | 31.4 | 23.2 | 23.6 | 22.9 |

Cylindrical | 52.6 | 47.3 | 42.4 | 39.2 | 33.2 | 25.2 | 17.0 | 16.4 |

VDW | 54.4 | 48.1 | 42.3 | 38.2 | 31.4 | 22.9 | 22.1 | 18.1 |

Geometric | 40.4 | 37.0 | 34.1 | 32.4 | 28.6 | 23.0 | 22.8 | 22.6 |

Redlich–Kwong | 88.8 | 78.5 | 69.1 | 62.3 | 51.3 | 37.3 | 33.9 | 26.6 |

Spherical | 148.8 | 127.9 | 109.1 | 95.0 | 74.7 | 51.4 | 48.5 | 37.5 |

Hamieh | 80.6 | 69.3 | 59.2 | 51.6 | 40.9 | 21.2 | 20.4 | 18.1 |

Dorris–Gray | 59.8 | 54.8 | 50.9 | 50.6 | 46.8 | 42.8 | 42.2 | 41.1 |

Hamieh–Gray | 105.6 | 88.7 | 74.9 | 67.0 | 55.0 | 44.1 | 37.2 | 30.2 |

Global average | 76.0 | 66.5 | 58.2 | 52.7 | 43.7 | 32.3 | 29.7 | 25.9 |

**Table 6.**Equations ${\gamma}_{s}^{d}\left(T\right)$ of alumina particles for various molecular models of n-alkanes, the dispersive surface entropy ${\epsilon}_{s}^{d}$, the extrapolated values ${\gamma}_{s}^{d}\left(T=0K\right)$ and the maximum of temperature ${T}_{Max}$.

Molecular Model | ${\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathit{T}\right)$ (mJ/m ^{2}) | ${\mathit{\epsilon}}_{\mathit{s}}^{\mathit{d}}=\mathit{d}{\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}/\mathit{d}\mathit{T}$ (mJ m ^{−2} K^{−1}) | ${\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathit{T}=0\mathit{K}\right)$ (mJ/m ^{2}) | ${\mathit{T}}_{\mathit{M}\mathit{a}\mathit{x}}\left(\mathit{K}\right)$ |
---|---|---|---|---|

Kiselev | ${\gamma}_{s}^{d}\left(T\right)$ = −0.232 T + 126.4 | −0.232 | 126.4 | 544.36 |

Cylindrical | ${\gamma}_{s}^{d}\left(T\right)$ = −0.275 T + 142.3 | −0.275 | 142.3 | 517.41 |

VDW | ${\gamma}_{s}^{d}\left(T\right)$ = −0.2674 T + 139.8 | −0.267 | 139.8 | 522.89 |

Geometric | ${\gamma}_{s}^{d}\left(T\right)$ = −0.138 T + 84.6 | −0.139 | 84.6 | 610.58 |

Redlich–Kwong | ${\gamma}_{s}^{d}\left(T\right)$ = −0.455 T + 235.1 | −0.456 | 235.1 | 516.05 |

Spherical | ${\gamma}_{s}^{d}\left(T\right)$ = −0.815 T + 407.2 | −0.815 | 407.2 | 499.39 |

Hamieh model | ${\gamma}_{s}^{d}\left(T\right)$ = 0.480 T + 233.9 | −0.480 | 233.9 | 487.21 |

Dorris–Gray | ${\gamma}_{s}^{d}\left(T\right)$ = −0.132 T + 100.7 | −0.133 | 100.7 | 760.08 |

Hamieh–Gray | ${\gamma}_{s}^{d}\left(T\right)$ = −0.500 T + 271.0 | −0.530 | 271.0 | 511.78 |

Global average | ${\gamma}_{s}^{d}\left(T\right)$ = −0.370 T + 141.2 | −0.370 | 193.4 | 523.42 |

**Table 7.**The linear equations of $-\Delta {G}_{a}^{sp}\left(T\right)$ (kJ/mol) of the polar solvents adsorbed on alumina particles as a function of the temperature T (K) for all models and methods.

Model or Method | Polar Solvent | $\mathit{Equation}-\mathbf{\Delta}{\mathit{G}}_{\mathit{a}}^{\mathit{s}\mathit{p}}\left(\mathit{T}\right)(\mathbf{kJ}/\mathbf{mol})$ |
---|---|---|

Kiselev | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.015 T + 9.951 |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.0950 T + 66.196 | |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.099 T + 49.816 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.104 T + 76.237 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.073 T + 55.663 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.053 T + 27.836 | |

Spherical | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.022 T + 12.846 |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.109 T + 73.138 | |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.115 T + 58.418 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.090 T + 71.08 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.095 T + 65.952 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.071 T + 37.748 | |

Geometric | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.038 T + 22.904 |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.119 T + 77.161 | |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.121 T + 60.18 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.081 T + 65.801 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.080 T + 59.029 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.060 T + 32.654 | |

Van der Waals (VDW) | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.017 T + 10.919 |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.101 T + 69.275 | |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.111 T + 56.128 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.085 T + 68.404 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.091 T + 64.627 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.066 T + 35.609 | |

Redlich–Kwong (R-K) | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.021 T + 12.349 |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.105 T + 70.824 | |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.113 T + 57.257 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.090 T + 70.46 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.096 T + 66.356 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.068 T + 36.511 | |

Cylindrical | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.021 T + 12.489 |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.135 T + 83.700 | |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.136 T + 65.871 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.088 T + 68.367 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.071 T + 53.71 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$= −0.041 T + 21.91 | |

Hamieh model | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.006 T + 8.164 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.007 T + 29.475 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.099 T + 51.024 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.110 T + 76.509 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.064 T + 56.551 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.035 T + 18.456 | |

Topological index | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.019 T + 19.115 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.111 T + 77.995 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.112 T + 58.858 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.088 T + 68.894 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.084 T + 64.482 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.041 T + 29.895 | |

Deformation polarizability | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = -0.022 T + 21.723 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = -0.083 T + 57.101 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.100 T + 50.004 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.0922 T + 71.692 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.092 T + 70.019 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.041 T + 29.774 | |

Vapor pressure | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = 0.001 T + 4.7609 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.087 T + 61.958 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.091 T + 43.784 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.084 T + 66.903 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.079 T + 59.071 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.033 T + 23.369 | |

Boiling point | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = 0.002 T + 4.0546 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.091 T + 63.571 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.089 T + 42.024 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.087 T + 68.002 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.075 T + 57.849 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.031 T + 22.645 | |

Enthalpy of vaporization ΔHvap(298K) | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = 0.001 T + 4.8875 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.086 T + 59.17 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.091 T + 43.106 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.086 T + 66.757 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.074 T + 56.843 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.033 T + 23.885 | |

Thermic enthalpy of vaporization ΔHvap(T) | CCl_{4} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.018 T + 10.116 |

CH_{2}Cl_{2} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.096 T + 62.393 | |

CHCl_{3} | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.115 T + 49.546 | |

Diethyl ether | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.112 T + 73.958 | |

THF | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.095 T + 62.454 | |

Toluene | $-\Delta {G}_{a}^{sp}\left(T\right)$ = −0.037 T + 25.095 |

**Table 8.**Values of the specific enthalpy ($-\Delta {H}_{a}^{sp}in{\mathrm{J}\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$) of the various polar solvents adsorbed on alumina by using the various molecular models, Hamieh model, topological index, deformation polarizability and vapor pressure methods compared to the global average with the standard deviation and the error percentage.

Probes | CCl_{4} | CHCl_{3} | CH_{2}Cl_{2} | Diethyl Ether | THF | Toluene |
---|---|---|---|---|---|---|

Kiselev | 9.95 | 49.82 | 66.20 | 76.24 | 55.66 | 27.84 |

Spherical | 12.85 | 58.42 | 73.14 | 71.08 | 65.95 | 37.75 |

Geometric | 22.90 | 60.18 | 77.16 | 65.80 | 59.03 | 32.65 |

VDW | 10.92 | 56.13 | 69.28 | 68.40 | 64.63 | 35.61 |

R-K | 12.35 | 57.26 | 70.82 | 70.46 | 66.36 | 36.51 |

Cylindrical | 12.49 | 65.87 | 83.70 | 68.37 | 53.71 | 21.91 |

Hamieh model | 8.16 | 51.02 | 29.48 | 76.51 | 56.55 | 18.46 |

Topological index | 19.12 | 58.86 | 78.00 | 68.89 | 64.48 | 29.90 |

Deformation polarizability | 21.72 | 50.00 | 57.10 | 71.69 | 70.02 | 29.77 |

Vapor pressure | 4.76 | 43.78 | 61.96 | 66.90 | 59.07 | 23.37 |

Boiling point | 4.05 | 42.02 | 63.57 | 68.00 | 57.85 | 22.65 |

ΔHvap(298K) | 4.89 | 43.11 | 59.17 | 66.76 | 56.84 | 23.89 |

ΔHvap(T) | 10.12 | 49.55 | 62.39 | 73.96 | 62.45 | 25.10 |

Average values | 11.87 | 52.77 | 65.54 | 70.24 | 60.97 | 28.11 |

Standard deviation | 6.16 | 7.34 | 13.41 | 3.52 | 4.99 | 6.17 |

Error percentage | 51.86 | 13.91 | 20.47 | 5.01 | 8.18 | 21.96 |

**Table 9.**Values of the specific entropy ($-\Delta {S}_{a}^{sp}in{\mathrm{J}\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$) of the various polar solvents adsorbed on alumina by using the various molecular models, Hamieh model, topological index, deformation polarizability and vapor pressure methods, compared to global average with the standard deviation and the error percentage.

Probes | CCl_{4} | CHCl_{3} | CH_{2}Cl_{2} | Diethyl Ether | THF | Toluene |
---|---|---|---|---|---|---|

Kiselev | 15.1 | 98.8 | 94.9 | 104.1 | 73.3 | 53 |

Spherical | 22.2 | 114.5 | 108.8 | 90.2 | 95.2 | 71.3 |

Geometric | 37.7 | 120.8 | 119.1 | 80.8 | 80.2 | 60.1 |

VDW | 17 | 111.1 | 101.1 | 84.9 | 91.4 | 65.6 |

R-K | 20.5 | 113.3 | 104.9 | 90.1 | 95.7 | 67.7 |

Cylindrical | 20.5 | 133.5 | 134.7 | 87.7 | 70.8 | 40.6 |

Hamieh model | 5.9 | 98.7 | 6.7 | 110.1 | 64.1 | 35 |

Topological index | 18.5 | 112 | 111.2 | 88.4 | 84.2 | 41.2 |

Deformation polarizability | 22 | 99.9 | 82.7 | 92.2 | 91.8 | 41.1 |

Vapor pressure | −0.6 | 91.2 | 86.6 | 83.7 | 78.8 | 32.8 |

Boiling point | −2.2 | 88.9 | 91.3 | 87 | 75.1 | 31.4 |

ΔHvap(298K) | −1 | 90.5 | 85.5 | 85.5 | 73.8 | 33 |

ΔHvap(T) | 18.1 | 114.7 | 96.1 | 111.8 | 95 | 37.3 |

Average values | 14.9 | 106.8 | 94.1 | 92.0 | 82.3 | 46.9 |

Standard deviation | 11.48 | 13.31 | 30.08 | 10.07 | 10.68 | 14.61 |

Error percentage | 77.02 | 12.46 | 31.97 | 10.94 | 12.98 | 31.15 |

**Table 10.**Values of the enthalpic acid base constants, ${K}_{A}$ and ${K}_{D}$(unitless), and the entropic acid base constants, ${\omega}_{A}$ and ${\omega}_{D}$(unitless), of alumina surface and the acid base ratios for the different used molecular models and IGC methods.

Models and IGC Methods | ${\mathit{K}}_{\mathit{A}}$ | ${\mathit{K}}_{\mathit{D}}$ | ${\mathit{K}}_{\mathit{D}}$$/{\mathit{K}}_{\mathit{A}}$ | ${10}^{3}\xb7{\mathit{\omega}}_{\mathit{A}}$ | ${10}^{3}\xb7{\mathit{\omega}}_{\mathit{D}}$ | ${\mathit{\omega}}_{\mathit{D}}$$/{\mathit{\omega}}_{\mathit{A}}$ |
---|---|---|---|---|---|---|

Kiselev | 0.578 | 2.705 | 4.68 | 0.72 | 4.71 | 6.5 |

Spherical | 0.665 | 3.093 | 4.65 | 0.91 | 5.42 | 6.0 |

Geometric | 0.553 | 3.676 | 6.65 | 0.68 | 6.34 | 9.3 |

VDW | 0.659 | 2.818 | 4.28 | 0.89 | 4.76 | 5.4 |

R-K | 0.674 | 2.961 | 4.40 | 0.92 | 5.11 | 5.5 |

Cylindrical | 0.534 | 2.879 | 5.39 | 0.64 | 5.09 | 7.9 |

Hamieh model | 0.624 | 1.831 | 2.93 | 0.72 | 2.79 | 3.9 |

Topological index | 0.633 | 3.250 | 5.13 | 0.82 | 4.27 | 5.2 |

Deformation polarizability | 0.705 | 3.034 | 4.30 | 0.92 | 3.97 | 4.3 |

Vapor pressure | 0.637 | 1.887 | 2.96 | 0.85 | 2.35 | 2.8 |

Boiling point | 0.626 | 1.863 | 2.97 | 0.82 | 2.36 | 2.9 |

DHvap | 0.612 | 1.928 | 3.15 | 0.80 | 2.46 | 3.1 |

DHvap(T) | 0.659 | 2.376 | 3.60 | 0.98 | 4.07 | 4.2 |

Average values | 0.628 | 2.639 | 4.20 | 0.82 | 4.13 | 5.0 |

Standard deviation | 0.05 | 0.61 | 0.10 | 1.29 | ||

Error percentage | 7.78 | 22.96 | 12.79 | 31.34 |

**Table 11.**London dispersive surface energy and enthalpic and entropic Lewis acid–base parameters of titania.

$\mathbf{Equation}{\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathit{T}\right)\mathbf{of}{\mathbf{TiO}}_{2}$ $\left(\mathbf{i}\mathbf{n}\mathbf{m}\mathbf{J}/{\mathbf{m}}^{2}\right),\mathit{T}\mathbf{in}\mathbf{K}$ | ${\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathit{T}\right)=-\mathbf{0.484}\mathit{T}+\mathbf{231.5}$ |
---|---|

${K}_{A}$ | 0.10 |

${K}_{D}$ | 0.97 |

${K}_{D}$/ | 9.72 |

${\omega}_{A}$ | 0.23 × 10^{−3} |

${\omega}_{D}$ | 2.71 × 10^{−3} |

${\omega}_{D}$/${\omega}_{A}$ | 11.60 |

$\mathbf{Equation}{\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathit{T}\right)\mathbf{of}{\mathbf{SiO}}_{2}$ $\left(\mathit{i}\mathit{n}\mathbf{m}\mathbf{J}/{\mathbf{m}}^{2}\right),\mathit{T}\mathbf{in}\mathbf{K}$ | ${\mathit{\gamma}}_{\mathit{s}}^{\mathit{d}}\left(\mathit{T}\right)=-0.99\mathit{T}+428$ |
---|---|

${K}_{A}$ | 0.23 |

${K}_{D}$ | 2.7 |

${K}_{D}$/${K}_{A}$ | 11.60 |

${\omega}_{A}$ | 1.21 × 10^{−3} |

${\omega}_{D}$ | −1.38 × 10^{−3} |

${\omega}_{D}$/${\omega}_{A}$ | −1.14 |

Parameter | Silica | Alumina | Titania |
---|---|---|---|

${\gamma}_{s}^{d}\left(T\right)$ of xide | ${\gamma}_{s}^{d}\left(T\right)=-0.99T+428$ | ${\gamma}_{s}^{d}\left(T\right)$ = 0.480T + 233.9 | ${\gamma}_{s}^{d}\left(T\right)=-0.484T+231.5$ |

${K}_{A}$ | 2.7 | 0.62 | 0.10 |

${K}_{D}$ | 0.23 | 1.83 | 0.97 |

${K}_{D}$/${K}_{A}$ | 0.09 | 2.93 | 9.72 |

${\omega}_{A}$ | 1.21 × 10^{−3} | 0.72 × 10^{−3} | 0.23 × 10^{−3} |

${\omega}_{D}$ | −1.38 × 10^{−3} | 2.79 × 10^{−3} | 2.71 × 10^{−3} |

${\omega}_{D}$/${\omega}_{A}$ | −1.14 | 3.9 | 11.60 |

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**MDPI and ACS Style**

Hamieh, T.
New Physicochemical Methodology for the Determination of the Surface Thermodynamic Properties of Solid Particles. *AppliedChem* **2023**, *3*, 229-255.
https://doi.org/10.3390/appliedchem3020015

**AMA Style**

Hamieh T.
New Physicochemical Methodology for the Determination of the Surface Thermodynamic Properties of Solid Particles. *AppliedChem*. 2023; 3(2):229-255.
https://doi.org/10.3390/appliedchem3020015

**Chicago/Turabian Style**

Hamieh, Tayssir.
2023. "New Physicochemical Methodology for the Determination of the Surface Thermodynamic Properties of Solid Particles" *AppliedChem* 3, no. 2: 229-255.
https://doi.org/10.3390/appliedchem3020015